Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of log-returns

The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)

vmr vhr vmrl pmr phr mmr mhr vmr_phr vhr_pmr
Min. : 0.868 0.849 0.801 0.904 0.878 0.988 0.977 0.979 0.967
1st Qu.: 1.044 1.039 1.013 1.042 1.068 1.013 1.013 1.021 1.011
Median : 1.097 1.099 1.085 1.084 1.128 1.085 1.113 1.102 1.094
Mean : 1.067 1.080 1.057 1.063 1.089 1.064 1.085 1.079 1.072
3rd Qu.: 1.136 1.160 1.128 1.107 1.182 1.101 1.128 1.121 1.107
Max. : 1.168 1.214 1.193 1.141 1.208 1.133 1.207 1.178 1.163

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.089 phr 1.182 phr 1.214 vhr
0.979 vmr_phr 1.044 vmr 1.113 mhr 1.085 mhr 1.160 vhr 1.208 phr
0.977 mhr 1.042 pmr 1.102 vmr_phr 1.080 vhr 1.136 vmr 1.207 mhr
0.967 vhr_pmr 1.039 vhr 1.099 vhr 1.079 vmr_phr 1.128 vmrl 1.193 vmrl
0.904 pmr 1.021 vmr_phr 1.097 vmr 1.072 vhr_pmr 1.128 mhr 1.178 vmr_phr
0.878 phr 1.013 vmrl 1.094 vhr_pmr 1.067 vmr 1.121 vmr_phr 1.168 vmr
0.868 vmr 1.013 mmr 1.085 vmrl 1.064 mmr 1.107 pmr 1.163 vhr_pmr
0.849 vhr 1.013 mhr 1.085 mmr 1.063 pmr 1.107 vhr_pmr 1.141 pmr
0.801 vmrl 1.011 vhr_pmr 1.084 pmr 1.057 vmrl 1.101 mmr 1.133 mmr

Correlations and covariance

Correlations

vmr vhr pmr phr
vmr 1.000 0.993 -0.197 -0.095
vhr 0.993 1.000 -0.119 -0.016
pmr -0.197 -0.119 1.000 0.957
phr -0.095 -0.016 0.957 1.000

Covariances

vmr vhr pmr phr
vmr 0.007 0.009 -0.001 -0.001
vhr 0.009 0.011 -0.001 0.000
pmr -0.001 -0.001 0.004 0.007
phr -0.001 0.000 0.007 0.011

Compare pension plans

Risk of loss

Risk of loss at least as big as x percent for a single period (year).
x values are row names.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 21.167 21.333 11.833 14.000 12.333 12.667 16.667 16.000
5 12.167 13.167 5.667 8.333 5.833 3.833 8.667 8.167
10 7.000 8.000 3.000 5.000 2.833 0.500 4.333 4.167
25 1.333 1.500 0.500 1.000 0.333 0.000 0.333 0.333
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 17.333 20.333 8.833 26.667 11.667 14.333 13.500 15.000
5 7.667 10.333 4.333 14.500 1.000 3.500 2.667 2.833
10 3.000 4.667 2.333 6.333 0.000 0.167 0.000 0.000
25 0.000 0.000 0.333 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 21.167 21.667 16.500 19.667 9.333 12.0 10.833 12.000
5 7.333 9.500 3.333 8.500 0.500 2.5 1.667 1.833
10 1.500 2.833 0.000 2.667 0.000 0.0 0.000 0.000
25 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000

Worst ranking for loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.333 vhr 13.167 vhr 8.000 vhr 1.500 vhr 0 vmr 0 vmr 0 vmr
21.167 vmr 12.167 vmr 7.000 vmr 1.333 vmr 0 vhr 0 vhr 0 vhr
16.667 vmr_phr 8.667 vmr_phr 5.000 phr 1.000 phr 0 pmr 0 pmr 0 pmr
16.000 vhr_pmr 8.333 phr 4.333 vmr_phr 0.500 pmr 0 phr 0 phr 0 phr
14.000 phr 8.167 vhr_pmr 4.167 vhr_pmr 0.333 mmr 0 mmr 0 mmr 0 mmr
12.667 mhr 5.833 mmr 3.000 pmr 0.333 vmr_phr 0 mhr 0 mhr 0 mhr
12.333 mmr 5.667 pmr 2.833 mmr 0.333 vhr_pmr 0 vmr_phr 0 vmr_phr 0 vmr_phr
11.833 pmr 3.833 mhr 0.500 mhr 0.000 mhr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
26.667 phr 14.500 phr 6.333 phr 0.333 pmr 0 vmr 0 vmr 0 vmr
20.333 vhr 10.333 vhr 4.667 vhr 0.000 vmr 0 vhr 0 vhr 0 vhr
17.333 vmr 7.667 vmr 3.000 vmr 0.000 vhr 0 pmr 0 pmr 0 pmr
15.000 vhr_pmr 4.333 pmr 2.333 pmr 0.000 phr 0 phr 0 phr 0 phr
14.333 mhr 3.500 mhr 0.167 mhr 0.000 mmr 0 mmr 0 mmr 0 mmr
13.500 vmr_phr 2.833 vhr_pmr 0.000 mmr 0.000 mhr 0 mhr 0 mhr 0 mhr
11.667 mmr 2.667 vmr_phr 0.000 vmr_phr 0.000 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
8.833 pmr 1.000 mmr 0.000 vhr_pmr 0.000 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.667 vhr 9.500 vhr 2.833 vhr 0 vmr 0 vmr 0 vmr 0 vmr
21.167 vmr 8.500 phr 2.667 phr 0 vhr 0 vhr 0 vhr 0 vhr
19.667 phr 7.333 vmr 1.500 vmr 0 pmr 0 pmr 0 pmr 0 pmr
16.500 pmr 3.333 pmr 0.000 pmr 0 phr 0 phr 0 phr 0 phr
12.000 mhr 2.500 mhr 0.000 mmr 0 mmr 0 mmr 0 mmr 0 mmr
12.000 vhr_pmr 1.833 vhr_pmr 0.000 mhr 0 mhr 0 mhr 0 mhr 0 mhr
10.833 vmr_phr 1.667 vmr_phr 0.000 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
9.333 mmr 0.500 mmr 0.000 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Chance of min gains

Chance of gains of at least x percent for a single period (year).
x values are row names.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 78.833 78.667 88.167 86.000 87.667 87.333 83.333 84.000
5 63.833 66.667 71.667 76.000 71.667 70.167 69.333 69.000
10 40.833 50.167 32.500 59.667 35.500 46.000 47.167 43.833
25 0.000 0.000 0.000 0.000 0.000 0.833 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 82.667 79.667 91.167 73.333 88.333 85.667 86.500 85.000
5 65.833 65.000 80.000 58.167 57.833 64.500 63.333 60.000
10 44.500 48.000 54.833 42.500 22.833 38.833 35.000 31.167
25 7.000 11.667 6.667 10.000 0.000 1.500 0.500 0.167
50 0.167 0.500 0.833 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 78.833 78.333 83.500 80.333 90.667 88.000 89.167 88.000
5 57.667 61.333 57.667 64.167 61.333 68.000 66.833 63.500
10 35.167 42.500 29.000 46.167 24.500 42.000 37.500 33.000
25 2.167 6.667 0.000 8.333 0.000 1.833 0.500 0.167
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Best ranking for gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
88.167 pmr 76.000 phr 59.667 phr 0.833 mhr 0 vmr 0 vmr
87.667 mmr 71.667 pmr 50.167 vhr 0.000 vmr 0 vhr 0 vhr
87.333 mhr 71.667 mmr 47.167 vmr_phr 0.000 vhr 0 pmr 0 pmr
86.000 phr 70.167 mhr 46.000 mhr 0.000 pmr 0 phr 0 phr
84.000 vhr_pmr 69.333 vmr_phr 43.833 vhr_pmr 0.000 phr 0 mmr 0 mmr
83.333 vmr_phr 69.000 vhr_pmr 40.833 vmr 0.000 mmr 0 mhr 0 mhr
78.833 vmr 66.667 vhr 35.500 mmr 0.000 vmr_phr 0 vmr_phr 0 vmr_phr
78.667 vhr 63.833 vmr 32.500 pmr 0.000 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
91.167 pmr 80.000 pmr 54.833 pmr 11.667 vhr 0.833 pmr 0 vmr
88.333 mmr 65.833 vmr 48.000 vhr 10.000 phr 0.500 vhr 0 vhr
86.500 vmr_phr 65.000 vhr 44.500 vmr 7.000 vmr 0.167 vmr 0 pmr
85.667 mhr 64.500 mhr 42.500 phr 6.667 pmr 0.000 phr 0 phr
85.000 vhr_pmr 63.333 vmr_phr 38.833 mhr 1.500 mhr 0.000 mmr 0 mmr
82.667 vmr 60.000 vhr_pmr 35.000 vmr_phr 0.500 vmr_phr 0.000 mhr 0 mhr
79.667 vhr 58.167 phr 31.167 vhr_pmr 0.167 vhr_pmr 0.000 vmr_phr 0 vmr_phr
73.333 phr 57.833 mmr 22.833 mmr 0.000 mmr 0.000 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
90.667 mmr 68.000 mhr 46.167 phr 8.333 phr 0 vmr 0 vmr
89.167 vmr_phr 66.833 vmr_phr 42.500 vhr 6.667 vhr 0 vhr 0 vhr
88.000 mhr 64.167 phr 42.000 mhr 2.167 vmr 0 pmr 0 pmr
88.000 vhr_pmr 63.500 vhr_pmr 37.500 vmr_phr 1.833 mhr 0 phr 0 phr
83.500 pmr 61.333 vhr 35.167 vmr 0.500 vmr_phr 0 mmr 0 mmr
80.333 phr 61.333 mmr 33.000 vhr_pmr 0.167 vhr_pmr 0 mhr 0 mhr
78.833 vmr 57.667 vmr 29.000 pmr 0.000 pmr 0 vmr_phr 0 vmr_phr
78.333 vhr 57.667 pmr 24.500 mmr 0.000 mmr 0 vhr_pmr 0 vhr_pmr

MC risk percentiles

Risk of loss at least as big as row name in percent from first to last period.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 4.94 2.74 2.00 1.08 0.29 0.05 0.16 0.11
5 4.28 2.34 1.86 0.97 0.23 0.03 0.15 0.08
10 3.75 2.03 1.66 0.81 0.16 0.01 0.13 0.04
25 2.24 1.28 1.29 0.47 0.10 0.01 0.09 0.01
50 0.89 0.41 0.75 0.23 0.01 0.00 0.00 0.00
90 0.05 0.01 0.23 0.02 0.00 0.00 0.00 0.00
99 0.00 0.00 0.07 0.00 0.00 0.00 0.00 0.00

1e6 sstd simulation paths of mhr:

0 5 10 25 50 90 99
prob_pct 0.118 0.095 0.076 0.036 0.008 0 0

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 0.06 0.09 0.82 0.32 0 0 0 0.02
5 0.05 0.08 0.78 0.23 0 0 0 0.02
10 0.05 0.03 0.77 0.21 0 0 0 0.01
25 0.01 0.00 0.63 0.07 0 0 0 0.00
50 0.00 0.00 0.46 0.00 0 0 0 0.00
90 0.00 0.00 0.14 0.00 0 0 0 0.00
99 0.00 0.00 0.04 0.00 0 0 0 0.00

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 0.04 0.03 0 0.03 0 0 0 0
5 0.03 0.01 0 0.02 0 0 0 0
10 0.01 0.01 0 0.01 0 0 0 0
25 0.00 0.01 0 0.00 0 0 0 0
50 0.00 0.00 0 0.00 0 0 0 0
90 0.00 0.00 0 0.00 0 0 0 0
99 0.00 0.00 0 0.00 0 0 0 0

Worst ranking for MC loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.94 vmr 4.28 vmr 3.75 vmr 2.24 vmr 0.89 vmr 0.23 pmr 0.07 pmr
2.74 vhr 2.34 vhr 2.03 vhr 1.29 pmr 0.75 pmr 0.05 vmr 0.00 vmr
2.00 pmr 1.86 pmr 1.66 pmr 1.28 vhr 0.41 vhr 0.02 phr 0.00 vhr
1.08 phr 0.97 phr 0.81 phr 0.47 phr 0.23 phr 0.01 vhr 0.00 phr
0.29 mmr 0.23 mmr 0.16 mmr 0.10 mmr 0.01 mmr 0.00 mmr 0.00 mmr
0.16 vmr_phr 0.15 vmr_phr 0.13 vmr_phr 0.09 vmr_phr 0.00 mhr 0.00 mhr 0.00 mhr
0.11 vhr_pmr 0.08 vhr_pmr 0.04 vhr_pmr 0.01 mhr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr
0.05 mhr 0.03 mhr 0.01 mhr 0.01 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
0.82 pmr 0.78 pmr 0.77 pmr 0.63 pmr 0.46 pmr 0.14 pmr 0.04 pmr
0.32 phr 0.23 phr 0.21 phr 0.07 phr 0.00 vmr 0.00 vmr 0.00 vmr
0.09 vhr 0.08 vhr 0.05 vmr 0.01 vmr 0.00 vhr 0.00 vhr 0.00 vhr
0.06 vmr 0.05 vmr 0.03 vhr 0.00 vhr 0.00 phr 0.00 phr 0.00 phr
0.02 vhr_pmr 0.02 vhr_pmr 0.01 vhr_pmr 0.00 mmr 0.00 mmr 0.00 mmr 0.00 mmr
0.00 mmr 0.00 mmr 0.00 mmr 0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr
0.00 mhr 0.00 mhr 0.00 mhr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr
0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
0.04 vmr 0.03 vmr 0.01 vmr 0.01 vhr 0 vmr 0 vmr 0 vmr
0.03 vhr 0.02 phr 0.01 vhr 0.00 vmr 0 vhr 0 vhr 0 vhr
0.03 phr 0.01 vhr 0.01 phr 0.00 pmr 0 pmr 0 pmr 0 pmr
0.00 pmr 0.00 pmr 0.00 pmr 0.00 phr 0 phr 0 phr 0 phr
0.00 mmr 0.00 mmr 0.00 mmr 0.00 mmr 0 mmr 0 mmr 0 mmr
0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr 0 mhr 0 mhr 0 mhr
0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

MC gains percentiles

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 95.06 97.26 98.00 98.92 99.71 99.95 99.84 99.89
5 94.32 96.90 97.74 98.79 99.62 99.94 99.77 99.87
10 93.61 96.43 97.57 98.68 99.53 99.93 99.71 99.84
25 91.02 94.88 96.82 98.36 99.14 99.80 99.51 99.61
50 85.77 91.56 94.85 97.57 97.74 99.37 98.83 98.87
100 72.15 83.27 88.04 94.65 90.32 97.44 96.09 94.53
200 40.32 61.23 59.24 84.78 49.18 86.21 78.99 65.32
300 16.58 39.46 23.29 70.32 11.45 63.67 50.32 29.21
400 5.42 22.99 4.80 54.11 1.09 38.72 24.53 9.21
500 1.49 12.41 0.58 38.19 0.08 19.04 9.33 2.32
1000 0.00 0.26 0.02 2.36 0.00 0.05 0.00 0.01

1e6 sstd simulation paths of mhr:

0 5 10 25 50 100 200 300 400 500 1000
prob 99.882 99.854 99.824 99.686 99.301 97.513 86.912 65.992 41.486 21.693 0.086

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 99.94 99.91 99.18 99.68 100.00 100.00 100.00 99.98
5 99.93 99.86 99.14 99.56 100.00 100.00 100.00 99.98
10 99.90 99.84 99.11 99.45 100.00 100.00 100.00 99.98
25 99.78 99.69 98.96 98.90 99.99 99.99 99.98 99.97
50 99.42 99.14 98.66 97.10 99.94 99.95 99.95 99.93
100 97.46 97.22 97.78 91.56 99.79 99.45 99.37 99.74
200 85.85 88.22 94.25 71.42 97.68 91.44 90.15 98.17
300 67.15 73.58 87.69 50.88 89.51 73.73 67.48 91.84
400 47.20 58.07 76.67 33.68 73.57 51.15 42.46 78.45
500 31.51 44.35 63.42 22.19 54.01 32.42 24.06 60.16
1000 3.95 9.81 17.22 2.62 6.55 2.23 0.76 9.20

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 99.96 99.97 100.00 99.97 100.00 100.00 100.00 100.00
5 99.92 99.97 100.00 99.97 100.00 100.00 100.00 100.00
10 99.89 99.97 100.00 99.97 100.00 100.00 100.00 100.00
25 99.74 99.83 99.94 99.90 100.00 99.99 100.00 100.00
50 98.87 99.42 99.68 99.68 100.00 99.98 99.99 100.00
100 93.02 96.54 95.09 98.10 98.96 99.90 99.58 99.39
200 64.16 80.34 57.26 88.20 68.67 94.01 89.86 81.54
300 32.76 57.39 19.82 70.42 22.22 74.48 62.15 43.44
400 13.85 37.25 4.78 51.39 4.38 47.03 33.23 16.91
500 5.67 22.92 0.98 35.22 0.69 25.61 15.85 5.96
1000 0.03 1.58 0.00 3.92 0.01 0.52 0.26 0.06

Best ranking for MC gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
99.95 mhr 99.94 mhr 99.93 mhr 99.80 mhr 99.37 mhr 97.44 mhr
99.89 vhr_pmr 99.87 vhr_pmr 99.84 vhr_pmr 99.61 vhr_pmr 98.87 vhr_pmr 96.09 vmr_phr
99.84 vmr_phr 99.77 vmr_phr 99.71 vmr_phr 99.51 vmr_phr 98.83 vmr_phr 94.65 phr
99.71 mmr 99.62 mmr 99.53 mmr 99.14 mmr 97.74 mmr 94.53 vhr_pmr
98.92 phr 98.79 phr 98.68 phr 98.36 phr 97.57 phr 90.32 mmr
98.00 pmr 97.74 pmr 97.57 pmr 96.82 pmr 94.85 pmr 88.04 pmr
97.26 vhr 96.90 vhr 96.43 vhr 94.88 vhr 91.56 vhr 83.27 vhr
95.06 vmr 94.32 vmr 93.61 vmr 91.02 vmr 85.77 vmr 72.15 vmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
86.21 mhr 70.32 phr 54.11 phr 38.19 phr 2.36 phr
84.78 phr 63.67 mhr 38.72 mhr 19.04 mhr 0.26 vhr
78.99 vmr_phr 50.32 vmr_phr 24.53 vmr_phr 12.41 vhr 0.05 mhr
65.32 vhr_pmr 39.46 vhr 22.99 vhr 9.33 vmr_phr 0.02 pmr
61.23 vhr 29.21 vhr_pmr 9.21 vhr_pmr 2.32 vhr_pmr 0.01 vhr_pmr
59.24 pmr 23.29 pmr 5.42 vmr 1.49 vmr 0.00 vmr
49.18 mmr 16.58 vmr 4.80 pmr 0.58 pmr 0.00 mmr
40.32 vmr 11.45 mmr 1.09 mmr 0.08 mmr 0.00 vmr_phr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mmr 100.00 mmr 100.00 mmr 99.99 mmr 99.95 mhr 99.79 mmr
100.00 mhr 100.00 mhr 100.00 mhr 99.99 mhr 99.95 vmr_phr 99.74 vhr_pmr
100.00 vmr_phr 100.00 vmr_phr 100.00 vmr_phr 99.98 vmr_phr 99.94 mmr 99.45 mhr
99.98 vhr_pmr 99.98 vhr_pmr 99.98 vhr_pmr 99.97 vhr_pmr 99.93 vhr_pmr 99.37 vmr_phr
99.94 vmr 99.93 vmr 99.90 vmr 99.78 vmr 99.42 vmr 97.78 pmr
99.91 vhr 99.86 vhr 99.84 vhr 99.69 vhr 99.14 vhr 97.46 vmr
99.68 phr 99.56 phr 99.45 phr 98.96 pmr 98.66 pmr 97.22 vhr
99.18 pmr 99.14 pmr 99.11 pmr 98.90 phr 97.10 phr 91.56 phr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
98.17 vhr_pmr 91.84 vhr_pmr 78.45 vhr_pmr 63.42 pmr 17.22 pmr
97.68 mmr 89.51 mmr 76.67 pmr 60.16 vhr_pmr 9.81 vhr
94.25 pmr 87.69 pmr 73.57 mmr 54.01 mmr 9.20 vhr_pmr
91.44 mhr 73.73 mhr 58.07 vhr 44.35 vhr 6.55 mmr
90.15 vmr_phr 73.58 vhr 51.15 mhr 32.42 mhr 3.95 vmr
88.22 vhr 67.48 vmr_phr 47.20 vmr 31.51 vmr 2.62 phr
85.85 vmr 67.15 vmr 42.46 vmr_phr 24.06 vmr_phr 2.23 mhr
71.42 phr 50.88 phr 33.68 phr 22.19 phr 0.76 vmr_phr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 pmr 100.00 pmr 100.00 pmr 100.00 mmr 100.00 mmr 99.90 mhr
100.00 mmr 100.00 mmr 100.00 mmr 100.00 vmr_phr 100.00 vhr_pmr 99.58 vmr_phr
100.00 mhr 100.00 mhr 100.00 mhr 100.00 vhr_pmr 99.99 vmr_phr 99.39 vhr_pmr
100.00 vmr_phr 100.00 vmr_phr 100.00 vmr_phr 99.99 mhr 99.98 mhr 98.96 mmr
100.00 vhr_pmr 100.00 vhr_pmr 100.00 vhr_pmr 99.94 pmr 99.68 pmr 98.10 phr
99.97 vhr 99.97 vhr 99.97 vhr 99.90 phr 99.68 phr 96.54 vhr
99.97 phr 99.97 phr 99.97 phr 99.83 vhr 99.42 vhr 95.09 pmr
99.96 vmr 99.92 vmr 99.89 vmr 99.74 vmr 98.87 vmr 93.02 vmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
94.01 mhr 74.48 mhr 51.39 phr 35.22 phr 3.92 phr
89.86 vmr_phr 70.42 phr 47.03 mhr 25.61 mhr 1.58 vhr
88.20 phr 62.15 vmr_phr 37.25 vhr 22.92 vhr 0.52 mhr
81.54 vhr_pmr 57.39 vhr 33.23 vmr_phr 15.85 vmr_phr 0.26 vmr_phr
80.34 vhr 43.44 vhr_pmr 16.91 vhr_pmr 5.96 vhr_pmr 0.06 vhr_pmr
68.67 mmr 32.76 vmr 13.85 vmr 5.67 vmr 0.03 vmr
64.16 vmr 22.22 mmr 4.78 pmr 0.98 pmr 0.01 mmr
57.26 pmr 19.82 pmr 4.38 mmr 0.69 mmr 0.00 pmr

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.048 0.063 0.058 0.084 0.059 0.082 0.067 0.062
s 0.120 0.126 0.123 0.121 0.088 0.071 0.091 0.090
nu 3.304 4.390 2.265 3.185 2.773 89.863 4.660 3.892
xi 0.034 0.019 0.477 0.018 0.029 0.770 0.048 0.019
R^2 0.993 0.995 0.991 0.964 0.890 0.961 0.927 0.933

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.084 0.090 0.102 0.073 0.058 0.075 0.071 0.065
s 0.106 0.122 0.345 0.119 0.050 0.071 0.065 0.063
nu 4.844 7.368 2.045 5682540.710 5283545.362 15657038.400 2680674.834 7710686.839
R^2 0.935 0.955 0.918 0.923 0.960 0.965 0.969 0.972

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.064 0.077 0.061 0.085 0.062 0.081 0.076 0.069
s 0.081 0.099 0.063 0.101 0.048 0.070 0.062 0.060
R^2 0.933 0.954 0.916 0.923 0.960 0.965 0.969 0.972

AIC and BIC

AIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -27.850 -21.575 -33.230 -23.726 -36.960 -24.261 -29.651 -31.100
std -16.385 -11.623 -22.924 -11.324 -33.923 -24.564 -27.112 -27.818
normal -20.316 -15.218 -27.005 -14.616 -34.127 -24.140 -27.388 -28.318

BIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -25.590 -19.315 -30.970 -21.466 -34.701 -22.001 -27.391 -28.841
std -14.125 -9.363 -20.664 -9.064 -31.663 -22.304 -24.852 -25.558
normal -18.056 -12.958 -24.746 -12.357 -31.867 -21.880 -25.129 -26.058

Fit statistics ranking

Skewed \(t\)-distribution (sstd):

m ranking s ranking R^2 ranking
0.084 phr 0.071 mhr 0.995 vhr
0.082 mhr 0.088 mmr 0.993 vmr
0.067 vmr_phr 0.090 vhr_pmr 0.991 pmr
0.063 vhr 0.091 vmr_phr 0.964 phr
0.062 vhr_pmr 0.120 vmr 0.961 mhr
0.059 mmr 0.121 phr 0.933 vhr_pmr
0.058 pmr 0.123 pmr 0.927 vmr_phr
0.048 vmr 0.126 vhr 0.890 mmr

Standardized \(t\)-distribution (std):

m ranking s ranking R^2 ranking
0.102 pmr 0.050 mmr 0.972 vhr_pmr
0.090 vhr 0.063 vhr_pmr 0.969 vmr_phr
0.084 vmr 0.065 vmr_phr 0.965 mhr
0.075 mhr 0.071 mhr 0.960 mmr
0.073 phr 0.106 vmr 0.955 vhr
0.071 vmr_phr 0.119 phr 0.935 vmr
0.065 vhr_pmr 0.122 vhr 0.923 phr
0.058 mmr 0.345 pmr 0.918 pmr

Normal distribution:

m ranking s ranking R^2 ranking
0.085 phr 0.048 mmr 0.972 vhr_pmr
0.081 mhr 0.060 vhr_pmr 0.969 vmr_phr
0.077 vhr 0.062 vmr_phr 0.965 mhr
0.076 vmr_phr 0.063 pmr 0.960 mmr
0.069 vhr_pmr 0.070 mhr 0.954 vhr
0.064 vmr 0.081 vmr 0.933 vmr
0.062 mmr 0.099 vhr 0.923 phr
0.061 pmr 0.101 phr 0.916 pmr

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 296.42 406.29 344.96 600.86 319.52 505.48 446.06 375.72
mc_s 134.29 210.50 119.95 274.68 88.20 172.23 151.43 120.31
mc_min 3.03 2.37 0.01 5.08 35.43 51.71 56.16 52.44
mc_max 915.54 1474.60 2824.80 1922.91 796.58 1326.58 1087.53 1319.52
dao_pct 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00
dai_pct 4.67 2.47 1.94 0.92 0.29 0.04 0.14 0.09

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 592.50 709.13 6.012997e+05 500.16 40290.88 597.63 544.36 3.552074e+26
mc_s 306.65 419.62 5.951009e+07 288.20 3902356.86 244.94 203.81 3.552074e+28
mc_min 74.74 90.15 1.000000e-02 63.30 117.24 125.28 131.47 8.999000e+01
mc_max 6365.75 5689.44 5.950808e+09 4376.28 390227286.46 2398.10 2311.76 3.552074e+30
dao_pct 0.00 0.00 2.000000e-02 0.00 0.00 0.00 0.00 0.000000e+00
dai_pct 0.04 0.02 8.100000e-01 0.27 0.00 0.00 0.00 1.000000e-02

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 387.99 517.74 349.54 610.41 368.48 559.96 501.75 431.08
mc_s 145.35 244.18 101.36 288.89 89.04 186.52 165.28 129.46
mc_min 91.52 71.65 106.12 72.91 139.12 148.34 142.53 151.90
mc_max 1442.08 3034.63 972.84 3328.70 1167.49 2199.87 1503.21 1373.78
dao_pct 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dai_pct 0.04 0.01 0.00 0.01 0.00 0.00 0.00 0.00

Ranking

Skewed \(t\)-distribution (sstd):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
600.86 phr 88.20 mmr 56.16 vmr_phr 2824.80 pmr 0.00 vmr 0.04 mhr
505.48 mhr 119.95 pmr 52.44 vhr_pmr 1922.91 phr 0.00 vhr 0.09 vhr_pmr
446.06 vmr_phr 120.31 vhr_pmr 51.71 mhr 1474.60 vhr 0.00 phr 0.14 vmr_phr
406.29 vhr 134.29 vmr 35.43 mmr 1326.58 mhr 0.00 mmr 0.29 mmr
375.72 vhr_pmr 151.43 vmr_phr 5.08 phr 1319.52 vhr_pmr 0.00 mhr 0.92 phr
344.96 pmr 172.23 mhr 3.03 vmr 1087.53 vmr_phr 0.00 vmr_phr 1.94 pmr
319.52 mmr 210.50 vhr 2.37 vhr 915.54 vmr 0.00 vhr_pmr 2.47 vhr
296.42 vmr 274.68 phr 0.01 pmr 796.58 mmr 0.01 pmr 4.67 vmr

Standardized \(t\)-distribution (std):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
3.552074e+26 vhr_pmr 2.038100e+02 vmr_phr 131.47 vmr_phr 3.552074e+30 vhr_pmr 0.00 vmr 0.00 mmr
6.012997e+05 pmr 2.449400e+02 mhr 125.28 mhr 5.950808e+09 pmr 0.00 vhr 0.00 mhr
4.029088e+04 mmr 2.882000e+02 phr 117.24 mmr 3.902273e+08 mmr 0.00 phr 0.00 vmr_phr
7.091300e+02 vhr 3.066500e+02 vmr 90.15 vhr 6.365750e+03 vmr 0.00 mmr 0.01 vhr_pmr
5.976300e+02 mhr 4.196200e+02 vhr 89.99 vhr_pmr 5.689440e+03 vhr 0.00 mhr 0.02 vhr
5.925000e+02 vmr 3.902357e+06 mmr 74.74 vmr 4.376280e+03 phr 0.00 vmr_phr 0.04 vmr
5.443600e+02 vmr_phr 5.951009e+07 pmr 63.30 phr 2.398100e+03 mhr 0.00 vhr_pmr 0.27 phr
5.001600e+02 phr 3.552074e+28 vhr_pmr 0.01 pmr 2.311760e+03 vmr_phr 0.02 pmr 0.81 pmr

Normal distribution:

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
610.41 phr 89.04 mmr 151.90 vhr_pmr 3328.70 phr 0 vmr 0.00 pmr
559.96 mhr 101.36 pmr 148.34 mhr 3034.63 vhr 0 vhr 0.00 mmr
517.74 vhr 129.46 vhr_pmr 142.53 vmr_phr 2199.87 mhr 0 pmr 0.00 mhr
501.75 vmr_phr 145.35 vmr 139.12 mmr 1503.21 vmr_phr 0 phr 0.00 vmr_phr
431.08 vhr_pmr 165.28 vmr_phr 106.12 pmr 1442.08 vmr 0 mmr 0.00 vhr_pmr
387.99 vmr 186.52 mhr 91.52 vmr 1373.78 vhr_pmr 0 mhr 0.01 vhr
368.48 mmr 244.18 vhr 72.91 phr 1167.49 mmr 0 vmr_phr 0.01 phr
349.54 pmr 288.89 phr 71.65 vhr 972.84 pmr 0 vhr_pmr 0.04 vmr

Compare Gaussian and skewed t-distribution fits

Gaussian fits

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
P_norm(X_min) 0.070 0.088 0.389 0.582 11.639 9.919 10.048 6.801
P_norm(X_max) 13.230 11.876 12.922 15.359 9.628 6.429 7.796 8.592
P_t(X_min) 5.377 5.080 3.489 4.315 10.570 8.015 13.008 10.520
P_t(X_max) 0.118 0.156 2.825 0.188 0.488 5.141 0.229 0.175

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
norm: avg yrs btw min 1438.131 1139.205 256.817 171.880 8.592 10.082 9.952 14.705
norm: avg yrs btw max 7.559 8.420 7.739 6.511 10.386 15.556 12.827 11.639
t: avg yrs btw min 18.596 19.687 28.663 23.173 9.461 12.476 7.688 9.506
t: avg yrs btw max 848.548 640.410 35.400 531.552 205.104 19.450 437.280 572.483

Lilliefors test

p-values for Lilliefors test.
Testing \(H_0\), that log-returns are Gaussian.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
p value 0.052 0.343 0.024 0.06 0.24 0.137 0.375 0.415

Wittgenstein’s Ruler

For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?

Conditional probabilities for smallest observed log-returns:

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \leq x_{\text{min}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.060 0.240 0.137 0.375 0.415
Prior prob 0.948 0.657 0.976 0.940 0.760 0.863 0.625 0.585
P[Gauss | Event] 0.661 0.223 0.854 0.859 0.642 0.844 0.433 0.362

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \geq x_{\text{max}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.060 0.240 0.137 0.375 0.415
Prior prob 0.948 0.657 0.976 0.940 0.760 0.863 0.625 0.585
P[Gauss | Event] 1.000 0.993 0.995 0.999 0.984 0.888 0.983 0.986

Velliv medium risk (vmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.2294983 0.3373312

Objective function plots

Velliv high risk (vhr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.5074609 0.4255322

Objective function plots

PFA medium risk (pmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.2936284 0.3062685

Objective function plots

PFA high risk (phr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.8379614 0.4397688

Objective function plots

Mix medium risk (mmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.1948623 0.2654885

Objective function plots

Mix high risk (mhr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.6413478 0.3380133

Objective function plots

Mix vmr+phr (vm_ph), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.5363616 0.3304634

Objective function plots

Mix vhr+pmr (mh_pm), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.3625460 0.3050122

Objective function plots

Comments

mhr has some nice properties:

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, under a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Many simulations of mc_mhr: num_paths = 1e6

1e6 paths:

Compare \(10^6\) and \(10^4\) paths for mhr:

mc_m mc_s mc_min mc_max dao_pct dai_pct
mc_mhr_1e6 505.90695 173.22176 21.09569 1734.83520 0.00000 0.07330
mc_mhr_1e4 505.47920 172.23152 51.70735 1326.58266 0.00000 0.04000
is_mhr_1e4 510.836 2331.167 205.398 232384.846 ibid. ibid.

Arithmetic vs geometric mean

Let \(m\) be the number of steps in each path and \(n\) be the number of paths. \(a\) is the initial capital. Use arithmetic mean for mean of all paths at time \(t\): \[\dfrac{a (e^{z_1} + e^{z_2} + \dots + e^{z_n})}{n}\] where \[z_i := x_{i, 1} + x_{i, 2} + \dots + x_{i, m}\] Use geometric mean for mean of all steps in a single path \(i\): \[a e^{\frac{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}{m}} = a \sqrt[m]{e^{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}}\]

So for Monte Carlo of returns after \(m\) periods, we

  • fit a skewed t-distribution to log-returns and use that distribution to simulate \(\{x_{i, j}\}_j^m\),
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • calculate the mean of \(\{z_i\}_i^n\):
    • \[\bar{z} = 100\dfrac{e^{z_1} + e^{z_2} + \dots + e^{z_n}}{n}\]

For Importance Sampling, we

  • model log-returns on a skewed t-distribution,
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • fit a skewed t-distribution to \(\{z_i\}_i^n\) and use it as our \(f\) density function from which we simulate \(\{h_i\}_i^n\),
    • In our case \(h\) and \(z\) are identical, because we have an idea for a distribution to simulate \(z\), but in general for IS \(h\) could be a function of \(z\).
  • calculate \(w* = \frac{f}{g^*}\), where \(g*\) is our proposal distribution, which minimizes the variance of \(h\cdot w\).
  • calculate the arithmetic mean of \(\{h_i w_i^{*}\}_i^n\):
    • \[100 \dfrac{e^{h_1 w_1^{*}} + e^{h_2 w_2^{*}} + \dots + e^{h_n w_n^{*}}}{n}\]

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.09406395 
## s(data_x): 0.4938257 
## m(data_y): 10.056 
## s(data_y): 2.546499 
## 
## m(data_x + data_y): 5.075032 
## s(data_x + data_y): 1.174483

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
101.882 101.336 5.604 5.097
101.632 101.360 5.779 5.307
101.414 101.172 5.974 5.177
101.555 101.892 5.845 5.217
101.406 101.454 5.877 5.138
101.432 101.415 5.995 5.170
101.842 101.370 5.586 5.247
101.864 101.568 5.662 5.232
101.363 101.686 5.752 5.228
101.097 101.407 5.851 5.257 
##       m_a             m_b             s_a             s_b       
##  Min.   :101.1   Min.   :101.2   Min.   :5.586   Min.   :5.097  
##  1st Qu.:101.4   1st Qu.:101.4   1st Qu.:5.685   1st Qu.:5.172  
##  Median :101.5   Median :101.4   Median :5.812   Median :5.223  
##  Mean   :101.5   Mean   :101.5   Mean   :5.792   Mean   :5.207  
##  3rd Qu.:101.8   3rd Qu.:101.5   3rd Qu.:5.870   3rd Qu.:5.244  
##  Max.   :101.9   Max.   :101.9   Max.   :5.995   Max.   :5.307

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05922   Min.   :0.05313  
##  1st Qu.:0.06581   1st Qu.:0.06282  
##  Median :0.06792   Median :0.06675  
##  Mean   :0.06812   Mean   :0.06991  
##  3rd Qu.:0.07040   3rd Qu.:0.07744  
##  Max.   :0.07812   Max.   :0.09388

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192